Hello, Boyrog!

I tried it myself and got into an awful mess . . .

So i was thinking about a geometry figure that is the set of points

which the sum of distances to *three* fixed points is a constant.

Does this geometry figure exist? I didn't bother with an equilateral triangle.

I used simpler coordinates. Code:

C|
o (0,a)
* | *
* | *
* | *
B * | * A
- - o - - - - + - - - - o - -
(-a,0) | (a,0)

We want point $\displaystyle P(x,y)$ so that: .$\displaystyle \overline{PA} + \overline{PB} + \overline{PC} \:=\:k$

So: .$\displaystyle \sqrt{(x-a)^2+y^2} + \sqrt{(x+a)^2+y^2} + \sqrt{x^2 + (y-a)^2} \:=\:k$

I see no practical way to eliminate the radicals.

Maybe *you* want to square the equation repeatedly.

. . I'll wait in the car . . .

.